In this study we have used a convergent and highly accurate mixed finite element technique to model the effect of fluid elasticity on the flow kinematics and the stress distribution in lid driven cavity flow. Our work is motivated by the desire to capture the important physical aspects of the basic flow and thus to better understand the purely elastic instability in recirculating flows which has been reported in the literature elsewhere [A.M. Grillet, E.S.G. Shaqfeh, Observations of viscoelastic instabilities in recirculation flows of Boger fluids, J. Non-Newtonian Fluid Mech. 64 (1996) 141-155; P. Pakdel, G.H. McKinley, Cavity flows of elastic liquids: purely elastic instablities, Phys. Fluids 10 (5) (1998) 1058-1070]. In our numerical investigations we have treated the corner singularities by incorporating a controlled amount of leakage which allows the computation of fully elastic mesh converged solutions. We begin by validating our Newtonian cavity results against previous work to show that the introduction of leakage does not appreciably modify the cavity recirculation flow. Then we examine the polymer stresses to understand how elasticity changes the flow kinematics, slowing the primary recirculation vortex and causing the vortex center to shift opposite of the direction of Lid motion. Variations of the cavity aspect ratio are also explored. Focusing on the corners we find that the leakage relieves the corner singularities and moreover, finite leakage helps explain the unusual behavior seen in the radial velocity in experiments. Finally, we have reexamined the previously proposed mechanisms for elastic instability in this flow and put forth a new instability mechanism. Together, these mechanisms may better explain the complex aspect ratio dependence of the onset of elastic instability in lid driven cavity flow.